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In [Phys. Rev. Lett. 113, 210501 (2014)], to achieve the optimal fixed-pointquantum search in the case of unknown fraction (denoted by $\lambda$) of targetitems, the analytical multiphase matching (AMPM) condition has been proposed.In this paper, we find out that the AMPM condition can also be used to designthe exact quantum search algorithm in the case of known $\lambda$, and theminimum number of iterations reaches the optimal level of existing exactalgorithms. Experiments are performed to demonstrate the proposed algorithm onIBM's quantum computer. In addition, we theoretically find two coincidentalphases with equal absolute value in our algorithm based on the AMPM conditionand that algorithm based on single-phase matching. Our work confirms thepracticability of the AMPM condition in the case of known $\lambda$, and ishelpful to understand the mechanism of this condition.

Exact quantum search based on analytical multiphase matching for known number of target items and the experimental demonstration on IBM Q